Proof (AQA A Level Maths: Pure): Exam Questions

Factorise .

Determine whether  is odd or even, where is a natural number.

Explain your answer clearly.

Use algebra to prove that the sum of any three consecutive even numbers is a multiple of 6.

Use algebra to prove that the square of an odd number is always odd.

Let be a natural number in the range .

Use proof by exhaustion to show that all possible values of differ from a multiple of 5 by 1.

"All integers of the form , where is a positive non-square integer less than 10, are prime."

Disprove this statement by means of a counter example.

Prove that

• if is odd, then is odd,

• if is even, then  is even.

Two non-zero rational numbers, and , are given by and  where , , and are non-zero integers with no common factors.

Determine whether

(i)  is rational

(ii) is rational

A function is given by

Show that for all real values of .

Use algebra to prove that the positive difference between a positive integer and its cube is the product of three consecutive integers.

Use algebra to prove that the sum of two rational numbers is rational.